BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 78, Number 4, July 1972
HECKE RINGS OF CONGRUENCE SUBGROUPS BY NELO D. ALLAN Communicated by M. Gerstenhaber, December 27,1971 Let A: be a p-adic field and let G be a reductive group defined over k. Let G be a semigroup in <#, i.e. a multiplicative subset with the same unity as G We shall assume that there exists an open compact subgroup A of à which is contained in G. Let âë(G9 A) be the free Z-module generated by the double cosets of G modulo A, with a product defined as in [3, Lemma 6]. We have an associative ring with unity which we shall call the Hecke Ring of G with respect to A. Let A0 be a normal subgroup of A satisfying our conditions H-l and H-2 of §1. Our purpose is to find generators and relations for 0t(G9 A0) = 0t. There exists a finitely generated polynomial ring Z\G~\ which together with the group ring Z[A/A0] generates 0t\ moreover 0t is a Z[A/A0]-bimodule having Z\p\ as basis. Our hypothesis H-l and H-2 are verified for the principal congruence subgroups of most of the classical groups considered in [2]. We thank Mr. J. Shalika for the helpful discussions during the prepara tion of this work and also for pointing out some hopes that this might bring in solving Harish-Chandra conjecture on the finite dimensionality of the irreducible continuous representations of these rings. 1. General results. Let J be a connected A:-closed subgroup of G con sisting only of semisimple elements, and N+ and N~ be maximal k- closed unipotent subgroups normalized by T. We set N+ = N+ n A, and U~ = TV" n A. We shall now state our first condition: Condition H-l. There exists a finitely generated semigroup D in T such that G = ADA (disjoint union), and for all del) we have dU+d~x c fZ+andrf-^-rfc: £/". We turn now to our second condition. We let A0 be a normal subgroup + of A and we set U£ = U n A0 and Uö = U~ n A0. We shall assume + that T N n A0 = (TnA0) • U%. Condition H-2. There exists a semigroup D in T such that A0 = UQ VUÖ for a certain subgroup V of A0 normalized by Z), and for all d in D we havedl/Jd"1 c l/Jandd^É/ödcz U^. Let us denote by 1 the unity of 0t and by g the double coset A0 gA0. We shall denote the product in 01 by *.
THEOREM 1. Condition H-2 implies that D = A0DA0 is a semigroup in G and 010, A0) ~ Z[D~]. A MS 1969 subject classifications. Primary 2220; Secondary 2265, 4256, 2070. Key words and phrases. Hecke rings, algebraic groups, locally compact groups, con volution algebras. Copyright © American Mathematical Society 1972 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 541 542 N. D. ALLAN [July
PROOF. Our condition implies that for all dl9d2e A we have A0d1A0d2A0 = A0d1d2A0, or dt * d2 = m • dxd2, with meZ, and it remains to prove that m = 1. From H-2 we can write
A0dA0 = (J {A0duj\j = 1,... ,w(d), use UÖ }. 1 Set Vj = dUjd" e N'. If A0^ = A0dx for some dx in Z>, then we have ijj = Ï and we may replace dx by d. This is equivalent to the existence of v e A0 such that vd = du,-. Now we set A0dtA0 = (J A0duf\ i = 1,2, and we recall that m is the number of pairs (/,;) such that 2 A0dxd2 = A03lW<.%Wj >. 1) 2) 1 We have vdxd2 = d1d2MÎ wj = d^h^d^ for someuj * in A0, and this implies v^ = 1 and consequently we have v\1} = I. Therefore m = 1. Q.E.D. THEOREM 2. TTie conditions H-l am/ H-2 wiï/i *te same Z) imply the finite generation as a ring of ai. Moreover 0t is a Z[A/A0]-bimodule having Z[D] as a basis.
PROOF. We let {du..., dr) be a set of generators for D. Let {at,..., LEMMA 1. af * 3* «r = ö,- * 3' * âs !ƒ and on/y (ƒ 3 = 3', ôtj e âf *L(3) ant/ 1 âs G /?'(3) * $ * âr, âf ^ * d = 3 * ^/or some ^ G A. 1 LEMMA 2. Suppose that for all the generators d of D we have dUöd' c U'.Ifd and d' are generators of D and if g e A, then cl*gcF = 0(3, g)* 61*3!* â', License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1972] HECKE RINGS OF CONGRUENCE SUBGROUPS 543 where a, a' € A and dxe D are such that Sgd' = a^a', and 0(d, g) = m • (sum of all elements ofL'(d) W, where Wis the subgroup ofL'(d) consisting of all â such that â * dg3! = ÏÏgd'). m is not greater than the order of the group g*L'(d!)*g-lnR(d). Finally, we would like to observe that M has an involution induced by g -• g"l in the case where G is a group. If, moreover, there exists 0 e A such that for all d e Z), d~ * = 0~id9, then the mapping â -• Öaö"1 induces the isomorphisms R'(d) *L'(d) and R(d) c*L(d). EXAMPLES. Let A' be a division algebra central over k9 O be the ring of integers of K, p its prime and n a fixed generator of p. Given a e K we shall denote by ord(a) the power of n in a. We let q be the number of elements in ©/p. For any positive integer m, 0/pm has qm elements. Let 5 be a subring of AT and let Mn(S) denote the ring of all n by n matrices with entries in 5; if g G Mn{S) and 1 ^ ij ^ n, then (g)lV will denote the (i, v)-entry of g and if we set (g)0- = g0, we write g = (g0); by etj we denote the matrix having 1 as (i,j)-entry and zero otherwise, and En or simply E will denote the identity of Mn(S). Gln(K) is the group of units of Mn(K). Case I. Ô = G/„(A). We let G = Gln(K)9 T = Tn = diagonal matrices in G, yv+(resp. TV") the group of all unipotent upper (resp. lower) triangular l n matrices in G, A = G0 = G/„(0). Let Dn = {deT\d = diag[V ,...,if \ c ear >*i è r2 ^ • • • è r„}. It is l ^at D„ satisfies H-l. For any r ^ 1 we set A0 r = Ar = {g e A|g = 1 mod p } = the rth congruence subgroup of A. We + have T iV nAr = (T n Ar) • üj. We let V = Jn Ar and V = Jo A. Condition H-2 will follow from the following lemma: LEMMA 3. Ar = IftVUZ = UjVU$. PROOF. Let g e Ar. As V normalizes both UQ and Uô we may assume that all diagonal entries of g are 1. If we consider g' = (E — ginein)g, i ^ w, then £ - ginein e U£ and (g'X-,, = 0. These operations will reduce to zero the nondiagonal entries of the last column of g. Now it suffices to transpose the resulting matrix, repeat the operation and apply induc tion. Q.E.D. REMARK. Let d e D be such that rn ^ 0. For any v eL'(d) we can choose l a representative such that d~ vd = u - (iiy)e U~ where utj = 0, for r i <;, uu = 1 for all i, and utj = ai}n with ord(a0) < r} - rt. Hence m r Also d_1 = örfö G = e and w(d) = q 9m = Xi License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 544 N. D. ALLAN [July Also if u is a unit, d *Z(u)wd = dZ(u)wd. This well determines the multi plication in ^. The case where G = Sln(k\ and D, A, Ar, V being the respective inter section of the corresponding groups with Sln{k\ is covered by our Theorem 2. Case II. Unitary groups. Let AT denote either k, or a quadratic extension of*, or else a quaternion division algebra over k. Let p denote respectively, the identity, the nontrivial automorphism of K over ft, and an involution of K. Clearly p can always be extended to an involution of Mn(K). Let h ^ 0 and let n = 2p + b; we subdivide every matrix geMn(K) into 9 blocks g = (gijl ij =1,2,3, in such way that g^g^e Mp(if) and g22eMb(K). Let D be the ring of integers of K9 and fix an HeMn(Ü), such that H" = yH,y = ±1,H = (fcyX*i3 = 7*31 = 0 = «iP + "" + *Pu h22 = V and htj = 0 otherwise, where V corresponds to an anisotropic form, if b # 0. We let G be the connected component of the group p {ge Gln(K)\g Hg = fi(g)H, where JJL is the multiplier}, and we let G0 be the correspondent group of V. We let x T= {heG\h = diag[z, h09 ju(/to)0(zT 0]9 h0 e G0 and z e Tp}, and we denote by N+, N~, A and A,, the intersection of the corresponding group in Gln(K) with G We let V= TnAr which clearly normalizes Ui and Uö. LEMMA 4. Ar = UQ VUQ. PROOF. Let g = (gtj) e Ar. We can apply Lemma 3 to g33 and we can _1 write g33 = n1h1u1. If we denote by n = diag[0(n?) 0, E, nj, h — 1 1 dmgl6(h{)- 0,E,h1] and u = diag[0(ti{)- e,E,ttil then nsU^heV, 1 1 1 and USUQ and replacinggby h~ n~ gu~ we may assume that g33 = E. p We take now £' = (£.,.) e £/J with g'12 = y6g 23V, g'13 = y0g?30 and an £23 = -£23 d #" = diag[£, /Î0, £] in V, for a convenient h0eG0; hence tftfgeUZ. QE.D. Now we consider A as in [2, §9], / = 0, D = {7rr|re A} and /)' = {de A|ju(d) = 1}. We take G = G and G' = {ge G|ju(g) = 1} and con sider their respective subgroups A, A', V, V\ etc. It can be easily checked that in all the cases discussed in [2, §9], our Theorems 1 and 2 remain valid for (G', A') and for (G, A) with the exception of the case (0)n = 2p. For G' we also have the extra assumptions of Lemma 2 and we also have a 1 0 = (0y); 013 = y031 = 0, 022 = E, 0y = 0 otherwise, such that d' = 0d0,7ip = 771. Closing this note we shall make two remarks: License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1972] HECKE RINGS OF CONGRUENCE SUBGROUPS 545 REMARK. For the adjoint representation of a Chevalley type group we have condition H-l by [1]. REMARK. Let Jf be a Hubert space and let ^(Jf ) be the algebra of all bounded operators on #?. Suppose that there exists a finite group G of unitary operators and a finite set of commuting operators Di9...,Dr all in â8{3tif) such that all D,'s are not necessarily normal. Let $& be the weak closure of the algebra generated by 1 and all the Dt. If we assume that every AeâS{J^) can be written as a finite sum of giByg^g^g^eG and Btj G âiï, does this necessarily imply that the dimension of J^ is finite? The positive answer of this question together with our Theorem 2 will imply Harish-Chandra's conjecture in these cases. REFERENCES 1. N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings ofp-adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math. No. 25 (1965), 5-48. MR 32 #2486. 2. I. Satake, Theory of spherical functions on reductive algebraic groups over p-adic fields, Inst. Hautes Études Sci. Publ. Math. No. 18 (1963), 5-69. MR 33 #4059. 3. T. Tamagawa, On the ^-function of a division algebra, Ann. of Math. (2) 77 (1963), 387-405. MR 26 #2468. UNIVERSIDAD NACIONAL DE COLOMBIA, BOGOTA, COLOMBIA, SOUTH AMERICA DEPARTMENT OF MATHEMATICS, UNIVERSITY OF WISCONSIN AT PARKSIDE, RACINE, WISCONSIN 53403 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use